论文标题
多个变量的双重数字的多项式函数
Polynomial functions over dual numbers of several variables
论文作者
论文摘要
令$ k \ in \ mathbb {n} \ setMinus \ {0 \} $。对于$ r $ $ r $的双数$ r $的双重戒指，$ r $的双重戒指是商ring $ r [x_1，\ ldots，x_k]/ i $，其中$ i $是集合$ \ {x_ix_j \ mid I，j = 1，j = 1，j = 1，j = 1，\ ldots，\ ldots，k \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \。该戒指可以被视为$ r [α_1，\ ldots，α_k] $，带有$α_iα_j= 0 $，其中$α_i= x_i+i $ for $ i，$ i，j = 1，\ ldots，k $。每当$ r $是有限的交换环时，我们都会研究$ r [α_1，\ ldots，α_k] $的多项式函数。我们根据$ r [α_1，\ ldots，α_k] $的多项式函数数量和多项式排列的数量得出计数公式，这取决于$ r [α_1，\ ldots $ r [α_1，\ ldots的多项式置换量的代数稳定器$ r $ r $的点稳定剂的顺序。此外，我们表明$ r $的稳定器组独立于变量$ k $的数量。此外，我们证明$ r [α_1，\ ldots，α_k] $上的功能$ f $是一个多项式函数。
Let $k\in \mathbb{N}\setminus\{0\}$. For a commutative ring $R$, the ring of dual numbers of $k$ variables over $R$ is the quotient ring $R[x_1,\ldots,x_k]/ I $, where $I$ is the ideal generated by the set $\{x_ix_j\mid i,j=1,\ldots,k\}$. This ring can be viewed as $R[α_1,\ldots,α_k]$ with $α_i α_j=0$, where $α_i=x_i+I$ for $i,j=1,\ldots,k$. We investigate the polynomial functions of $R[α_1,\ldots,α_k]$ whenever $R$ is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on $R[α_1,\ldots,α_k]$ depending on the order of the pointwise stabilizer of the subring of constants $R$ in the group of polynomial permutations of $R[α_1,\ldots,α_k]$. Further, we show that the stabilizer group of $R$ is independent of the number of variables $k$. Moreover, we prove that a function $F$ on $R[α_1,\ldots,α_k]$ is a polynomial function if and only if a system of linear equations on $R$ that depends on $F$ has a solution.
